Question

A torus has the following parametrisation: r(u,v)=((2+cosu)cosv,(2+cosu)sinv,sinu), where u,v∈[0,2π].

(a) Plot the torus using Python. See plotting tips below. Include only screenshot(s) - do not submit the ipynb file. Make sure the aspect ratio of your plot is roughly equal in all x,y,z directions so the surface looks like a doughnut and doesn't appear stretched. Label the axes. [ 3 marks]
(b) Using the Divergence Theorem with F(x,y,z)=(x,0,0), calculate the volume of the torus. [5 marks]

Answer 1

The question addresses the construction of a 3D plot of a torus using Python and the application of the Divergence Theorem to calculate the volume of the torus using a given vector field.

Step-by-step explanation:

The question falls within the field of mathematics, specifically focusing on geometry and vector calculus. Part (a) of the question deals with the visualization of a torus using Python, which would typically require the use of programming libraries such as matplotlib to generate a 3D plot of the parametric equations given for the torus. This part of the task would involve utilizing a meshgrid to create a grid of values for u and v, computing the corresponding x, y, and z coordinates, and then plotting them in 3D space to create the surface of the torus. The plot should be properly scaled so that the aspect ratio is equal in all directions, with labeled axes for clarity.

Part (b) of the question involves applying the Divergence Theorem to calculate the volume of the torus using a specific vector field F(x,y,z) = (x,0,0). This would require finding the divergence of the vector field and then integrating that divergence over the volume of the torus. As the parametrization of the torus is given, this task would involve converting the divergence to cylindrical coordinates, and then performing the integration over the appropriate bounds for u and v to determine the volume enclosed by the surface.